Problem 27 - Algebra 2

Question

Verify that each of the following is an identity. $$ \frac{1}{\sin x \cos x}-\frac{\cos x}{\sin x}=\tan x $$

Step-by-Step Solution

Answer

The given expression is an identity.

Step 1: Start with the Left Side

We will begin by simplifying the left side of the equation: \( \frac{1}{\sin x \cos x} - \frac{\cos x}{\sin x} \).

Step 2: Find a Common Denominator

Identify a common denominator for the fractions. The common denominator for \( \frac{1}{\sin x \cos x} \) and \( \frac{\cos x}{\sin x} \) is \( \sin x \cos x \).

Step 3: Express Each Term with the Common Denominator

Write each term over the common denominator:\[\frac{1}{\sin x \cos x} = \frac{1}{\sin x \cos x}\]\[\frac{\cos x}{\sin x} = \frac{\cos x \cos x}{\sin x \cos x} = \frac{\cos^2 x}{\sin x \cos x}\]

Step 4: Combine the Fractions

Subtract the second fraction from the first:\[\frac{1}{\sin x \cos x} - \frac{\cos^2 x}{\sin x \cos x} = \frac{1 - \cos^2 x}{\sin x \cos x}\]

Step 5: Simplify Using a Trigonometric Identity

Apply the Pythagorean identity \( \sin^2 x = 1 - \cos^2 x \) to the numerator:\[\frac{1 - \cos^2 x}{\sin x \cos x} = \frac{\sin^2 x}{\sin x \cos x}\]

Step 6: Simplify the Fraction

Cancel one \( \sin x \) from the numerator with the denominator:\[\frac{\sin^2 x}{\sin x \cos x} = \frac{\sin x}{\cos x} = \tan x\]

Step 7: Verify Identity

Since we have simplified the left side to \( \tan x \), which matches the right side of the equation, we have verified that the identity holds.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables they contain, representing the fundamental relationships between trigonometric functions. They simplify complex trigonometric expressions and are often used in calculus, physics, and engineering. Some key trigonometric identities include:

• Reciprocal Identities: These relate trigonometric functions to their reciprocals, such as \( \csc x = \frac{1}{\sin x} \).
• Quotient Identities: They express one trigonometric function as a fraction of others, like \( \tan x = \frac{\sin x}{\cos x} \).
• Pythagorean Identities: The most famous is \( \sin^2 x + \cos^2 x = 1 \).

To verify an identity, like the one given in the exercise, you need to manipulate the expression using these identities until both sides match. It involves simplifying both sides or converting one side to look like the other. By doing so, we confirm the equation's validity.

Common Denominator

Finding a common denominator is crucial when dealing with fractions because it allows you to combine them. A common denominator is essentially a shared multiple of the denominators you have. In the exercise, the fractions \( \frac{1}{\sin x \cos x} \) and \( \frac{\cos x}{\sin x} \) need a common denominator to be combined.
Here are steps to find and use a common denominator:

• Identify the denominators you have: \( \sin x \cos x \) and \( \sin x \).
• Determine the least common multiple, which is \( \sin x \cos x \) in this case.
• Express each fraction with this new denominator by adjusting the numerators accordingly.

This makes subtraction or addition straightforward, as shown in the original solution when combining the given fractions.

Pythagorean Identity

The Pythagorean identity is one of the core trigonometric identities that connects sine and cosine functions. It states that \( \sin^2 x + \cos^2 x = 1 \), and can take several forms depending on the situation. In the exercise, it is used to transform the expression \( 1 - \cos^2 x \) into \( \sin^2 x \).
This identity is derived from the Pythagorean theorem and is crucial in simplifying trigonometric expressions. Here's how it's applied in exercises:

• Use \( \sin^2 x = 1 - \cos^2 x \) if you have \( 1 - \cos^2 x \).
• Use \( \cos^2 x = 1 - \sin^2 x \) if you have \( 1 - \sin^2 x \).

The key is recognizing parts of your equation that fit into this identity, like the numerator in the problem, allowing for simplification and verification of identities.

Simplifying Fractions

Simplifying fractions is about making them as simple as possible while keeping their value the same. In trigonometry, this often involves canceling terms or reducing complex fractions. In the exercise, you simplify \( \frac{\sin^2 x}{\sin x \cos x} \) to \( \frac{\sin x}{\cos x} \) by canceling a \( \sin x \) from both the numerator and the denominator.
Here are steps to simplify fractions effectively:

• Look for common terms in the numerator and denominator that you can cancel.
• Use identities, like the Pythagorean identity, to convert expressions to simpler forms.
• Once simplified, compare the result to the original problem's other side to verify solutions, like equating \( \frac{\sin x}{\cos x} \) to \( \tan x \).

This process not only makes calculations easier but also helps in verifying complex trigonometric identities confidently.